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Mirrors > Home > MPE Home > Th. List > Mathboxes > un0.1 | Structured version Visualization version GIF version |
Description: ⊤ is the constant true, a tautology (see df-tru 1546). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
un0.1.1 | ⊢ ( ⊤ ▶ 𝜑 ) |
un0.1.2 | ⊢ ( 𝜓 ▶ 𝜒 ) |
un0.1.3 | ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) |
Ref | Expression |
---|---|
un0.1 | ⊢ ( 𝜓 ▶ 𝜃 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un0.1.1 | . . . 4 ⊢ ( ⊤ ▶ 𝜑 ) | |
2 | 1 | in1 41805 | . . 3 ⊢ (⊤ → 𝜑) |
3 | un0.1.2 | . . . 4 ⊢ ( 𝜓 ▶ 𝜒 ) | |
4 | 3 | in1 41805 | . . 3 ⊢ (𝜓 → 𝜒) |
5 | un0.1.3 | . . . 4 ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) | |
6 | 5 | dfvd2ani 41817 | . . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃) |
7 | 2, 4, 6 | uun0.1 42012 | . 2 ⊢ (𝜓 → 𝜃) |
8 | 7 | dfvd1ir 41807 | 1 ⊢ ( 𝜓 ▶ 𝜃 ) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1544 ( wvd1 41803 ( wvhc2 41814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-vd1 41804 df-vhc2 41815 |
This theorem is referenced by: sspwimpVD 42153 |
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